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Yukti Vault

Calculus

Definite Integration: JEE Main & Advanced PYQs

392 previous year questions on Definite Integration from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 351-392 of 392

  1. $\int_{0}^{\pi}[\cot x]\,dx$, where $[\,.\,]$ denotes the greatest integer function, is equal to :
  2. The value of $\int_{0}^{1}\frac{8\log(1+x)}{1+x^{2}}\,dx$ is
  3. Match the following: (i) $\displaystyle\int_0^{\pi/2}(\sin x)^{\cos x}\left(\cos x\cot x-\log(\sin x)^{\sin x}\right)dx$ (ii) Area bounded by $-4y^2=x$ and $x-1=-5y^2$ …
  4. Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate bubbles in the $4\times4$ matrix given in the ORS. …
  5. Show that $\displaystyle\int_{0}^{\pi}x\,f(\sin x)\,dx=\frac{\pi}{2}\int_{0}^{\pi}f(\sin x)\,dx$.
  6. Given a function $f(x)$ such that (i) it is integrable over every interval on the real line and (ii) $f(t+x)=f(x)$, for every $x$ and a real $t$, then show that the …
  7. Show that $\displaystyle\lim_{n\to\infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{6n}\right)=\log6$.
  8. If $f$ and $g$ are continuous functions on $[0,a]$ satisfying $f(x)=f(a-x)$ and $g(x)+g(a-x)=2$, then show that $\displaystyle\int_0^{a}f(x)g(x)\,dx=\int_0^{a}f(x)\,dx$.
  9. Let $a+b=4$, where $a 0$ for all $x$, prove that $\displaystyle\int_0^{a}g(x)\,dx+\int_0^{b}g(x)\,dx$ increases as $(b-a)$ increases.
  10. Show that $\displaystyle\int_{0}^{\pi/2}f(\sin2x)\sin x\,dx=\sqrt2\int_{0}^{\pi/4}f(\cos2x)\cos x\,dx$
  11. If $f$ is a continuous function with $\displaystyle\int_0^{x}f(t)\,dt\to\infty$ as $|x|\to\infty$, then show that every line $y=mx$ intersects the curve …
  12. Prove that for any positive integer $k$, $\dfrac{\sin2kx}{\sin x}=2\left[\cos x+\cos3x+\cdots+\cos(2k-1)x\right]$ Hence prove that …
  13. Let $I_m=\displaystyle\int_0^{\pi}\dfrac{1-\cos mx}{1-\cos x}\,dx$. Use mathematical induction to prove that $I_m=m\pi$, $m=0,1,2,\ldots$
  14. Show that $\displaystyle\int_{0}^{n\pi+v}|\sin x|\,dx=2n+1-\cos v$ where $n$ is a positive integer and $0\le v<\pi$.
  15. If $f$ is an even function then prove that $$\int_0^{\pi/2}f(\cos2x)\cos x\,dx=\sqrt2\int_0^{\pi/4}f(\sin2x)\cos x\,dx.$$
  16. If $u+v+w=91$, and $2u=3v=4w$, then the value of $\displaystyle\int_{u-v+w}^{2u+v+4w}e^{(x-[x])}dx$, where $[x]$ denotes the greatest integer $\le x$, is equal to
  17. The value of the integral $\displaystyle\int_{1}^{6}\dfrac{\log_{e}x\,dx}{\log_{e}x^{2}+\log_{e}\left(x^{2}-14x+49\right)}$ is
  18. Let $f:\mathbf{R}\to\mathbf{R}$ be a continuous function such that $f(x)+f(x+1)=2$, for all $x\in\mathbf{R}$. If $I_1=\displaystyle\int_{0}^{8}f(x)\,dx$ and …
  19. Let $f:(0,2)\to\mathbf{R}$ be defined as $f(x)=\log_2\left(1+\tan\left(\dfrac{\pi x}{4}\right)\right)$. Then …
  20. Let $f(x)=\begin{cases}-2, & -2\le x\le0\\ x-2, & 0<x\le2\end{cases}$ and $h(x)=f(|x|)+|f(x)|$. Then $\displaystyle\int_{-2}^{2}h(x)\,dx$ is equal to :
  21. If $\displaystyle\int_{0}^{\pi/4}\dfrac{\sin^{2}x}{1+\sin x\cos x}\,dx=\dfrac{1}{a}\log_e\left(\dfrac{a}{3}\right)+\dfrac{\pi}{b\sqrt{3}}$, where $a,b\in\mathbf{N}$ …
  22. If the value of the integral $\displaystyle\int_{-1}^{1}\dfrac{\cos\alpha x}{1+3^{x}}\,dx$ is $\dfrac{2}{\pi}$. Then, a value of $\alpha$ is
  23. The value of $\displaystyle\int_{-\pi}^{\pi}\dfrac{2y(1+\sin y)}{1+\cos^{2}y}\,dy$ is :
  24. The integral $\displaystyle\int_{0}^{\pi/4}\dfrac{136\sin x}{3\sin x+5\cos x}\,dx$ is equal to :
  25. If $f(t)=\displaystyle\int_{0}^{\pi}\dfrac{2x\,dx}{1-\cos^{2}t\sin^{2}x}$, $0<t<\pi$, then the value of $\displaystyle\int_{0}^{\pi/2}\dfrac{\pi^{2}\,dt}{f(t)}$ equals …
  26. Let $\beta(m,n)=\displaystyle\int_{0}^{1}x^{m-1}(1-x)^{n-1}dx$, $m,n>0$. If $\displaystyle\int_{0}^{1}\left(1-x^{10}\right)^{20}dx=a\times\beta(b,c)$, then $100(a+b+c)$ …
  27. If $I_1=\displaystyle\int_{0}^{1}\left(1-x^{50}\right)^{100}dx$ and $I_2=\displaystyle\int_{0}^{1}\left(1-x^{50}\right)^{101}dx$ such that $I_2=\alpha I_1$ then $\alpha$ …
  28. Let for $f(x)=7\tan^{8}x+7\tan^{6}x-3\tan^{4}x-3\tan^{2}x$, $I_1=\displaystyle\int_{0}^{\pi/4}f(x)\,dx$ and $I_2=\displaystyle\int_{0}^{\pi/4}xf(x)\,dx$. Then …
  29. The value of …
  30. Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\displaystyle\int_{0}^{x}tf(t)\,dt$. If $g\left(x^{3}\right)=x^{6}+x^{7}$ …
  31. Let $f:\mathbf{R}\to\mathbf{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=xf(x)$ for all $x\in\mathbf{R}$, $\displaystyle\int_{0}^{2}xF'(x)\,dx=6$ …
  32. $\lim\limits_{n\to\infty}\frac{1^{p}+2^{p}+3^{p}+\cdots+n^{p}}{n^{p+1}}$ is
  33. $\lim\limits_{n\to\infty}\frac{1+2^{4}+3^{4}+\cdots+n^{4}}{n^{5}}-\lim\limits_{n\to\infty}\frac{1+2^{3}+3^{3}+\cdots+n^{3}}{n^{5}}$
  34. $\lim\limits_{n\to\infty}\sum\limits_{r=1}^{n}\frac1ne^{\frac rn}$ is
  35. $\lim\limits_{n\to\infty}\left[\frac{1}{n^{2}}\sec^{2}\frac{1}{n^{2}}+\frac{2}{n^{2}}\sec^{2}\frac{4}{n^{2}}+\cdots+\frac1n\sec^{2}1\right]$ equals
  36. The integral $\int_{7\pi/4}^{7\pi/3}\sqrt{\tan^{2}x}\,dx$ is equal to :
  37. The value of $\int_0^\pi|\cos x|^3\,dx$ is
  38. Let $f(x)=\int_0^x g(t)\,dt$, where $g$ is a non-zero even function. If $f(x+5)=g(x)$, then $\int_0^x f(t)\,dt$ equals
  39. If $[t]$ denotes the greatest integer $\le t$, then the value of $\frac{3(e-1)}{e}\int_1^2x^2e^{[x]+[x^3]}\,dx$ is :
  40. If $f:\mathbf{R}\to\mathbf{R}$ is given by $f(x)=x+1$, then the value of …
  41. Let $g(t)=\displaystyle\int_{-\pi/2}^{\pi/2}\cos\left(\dfrac{\pi}{4}t+f(x)\right)dx$, where $f(x)=\log_e\left(x+\sqrt{x^2+1}\right)$, $x\in\mathbf{R}$. Then which one of …
  42. If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2}\left[[x]-\sin x\right]dx$ is equal to …

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